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Bertrand framed surfaces in the Euclidean 3-space and its applications

Nozomi Nakatsuyama, Masatomo Takahashi

TL;DR

This work extends Bertrand-type geometry from curves to framed surfaces in a Euclidean setting by introducing Bertrand framed surfaces and deriving their existence conditions across all mate configurations. It defines caustics and involutes directly for framed surfaces via moving frames and establishes when these operations invert each other, revealing a structured set of invariants $(\\mathcal{G},\\mathcal{F}_1,\\mathcal{F}_2)$ and curvature triplets $(J^F, K^F, H^F)$. A new tangential-direction framed surface operation is introduced, expanding the toolkit of frame-transformations and their impact on invariants. The paper also provides explicit transformation rules for basic invariants under these operations and illustrates the theory with detailed examples (including cuspidal edges and cross-caps), highlighting the role of frontals and singular geometry in the differential geometry of framed surfaces.

Abstract

A framed surface is a smooth surface in the Euclidean space with a moving frame. By using the moving frame, we can define Bertrand framed surfaces as the same idea as Bertrand framed curves. Then we find the caustics and involutes as Bertrand framed surfaces. As applications, we can directly define the caustics and involutes of framed surfaces, and give conditions that the caustics and involutes are inverse operations of framed surfaces like as those of Legendre curves. Moreover, a framed surface is one of the Bertrand framed surfaces if and only if another caustic of the involute exists, under conditions. Furthermore, we find a new such operation, the so-called tangential direction framed surfaces.

Bertrand framed surfaces in the Euclidean 3-space and its applications

TL;DR

This work extends Bertrand-type geometry from curves to framed surfaces in a Euclidean setting by introducing Bertrand framed surfaces and deriving their existence conditions across all mate configurations. It defines caustics and involutes directly for framed surfaces via moving frames and establishes when these operations invert each other, revealing a structured set of invariants and curvature triplets . A new tangential-direction framed surface operation is introduced, expanding the toolkit of frame-transformations and their impact on invariants. The paper also provides explicit transformation rules for basic invariants under these operations and illustrates the theory with detailed examples (including cuspidal edges and cross-caps), highlighting the role of frontals and singular geometry in the differential geometry of framed surfaces.

Abstract

A framed surface is a smooth surface in the Euclidean space with a moving frame. By using the moving frame, we can define Bertrand framed surfaces as the same idea as Bertrand framed curves. Then we find the caustics and involutes as Bertrand framed surfaces. As applications, we can directly define the caustics and involutes of framed surfaces, and give conditions that the caustics and involutes are inverse operations of framed surfaces like as those of Legendre curves. Moreover, a framed surface is one of the Bertrand framed surfaces if and only if another caustic of the involute exists, under conditions. Furthermore, we find a new such operation, the so-called tangential direction framed surfaces.
Paper Structure (6 sections, 31 theorems, 109 equations)

This paper contains 6 sections, 31 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bertrand framed surfaces
  4. Caustics and involutes of framed surfaces
  5. Tangential direction framed surfaces
  6. Examples

Key Result

Theorem 2.3

Let $U$ be a simply connected domain in $\mathbb{R}^2$ and let $a_i,b_i,e_i,f_i,g_i:U \to \mathbb{R}, i=1,2$ be smooth functions with the integrability conditions $()$. Then there exists a framed surface $(\hbox{\boldmath $x$},\hbox{\boldmath $n$},\hbox{\boldmath $s$}):U \to \mathbb{R}^3 \times \Del

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: The Existence Theorem for framed surfaces
  • Theorem 2.4: The Uniqueness Theorem for framed surfaces
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 56 more